Two Quantities Rising Together Is Not Enough to Be Proportional
Learn what direct variation really requires, find the constant of proportionality from any pair of values, solve problems by the unitary method and by proportion, and test a table of values properly.
Do two quantities that increase together always vary directly?
No — and this is the point the whole chapter turns on.
A taxi fare rises as the distance rises, but it is not in direct variation with distance, because there is usually a fixed charge before the meter starts. Double the distance and the fare does not double, since the fixed part was not doubled.
Direct variation demands something much stricter than rising together: the ratio of the two quantities must stay exactly the same. Doubling one must double the other, and halving one must halve the other.
So the test is arithmetical, not visual. This page covers the first part of the ICSE Class 8 Mathematics chapter on variation.
A taxi fare rises as the distance rises, but it is not in direct variation with distance, because there is usually a fixed charge before the meter starts. Double the distance and the fare does not double, since the fixed part was not doubled.
Direct variation demands something much stricter than rising together: the ratio of the two quantities must stay exactly the same. Doubling one must double the other, and halving one must halve the other.
So the test is arithmetical, not visual. This page covers the first part of the ICSE Class 8 Mathematics chapter on variation.
Formula
What is direct variation and how do you find the constant?
Two quantities are in direct variation if an increase in one produces a proportionate increase in the other, so that their ratio remains constant.
Written and read * varies directly as *, it means
where is the constant of proportionality, also called the constant of variation.
**Finding needs only one pair of values. Divide the second quantity by the first, and the answer holds for every other pair.
Worked example 1.** If pens cost , find the constant of proportionality and the cost of pens.
Notice what means here — it is the cost of one pen, and it carries the unit rupees per pen. In every direct variation the constant is a rate, which is why identifying it makes the problem concrete.
Worked example 2. A car travels on of fuel. How far will it travel on ?
Everyday pairs that do vary directly:
- Cost and the number of articles bought, at a fixed rate
- Distance and time, at a constant speed
- Weight and the number of identical objects
- The circumference of a circle and its diameter, with
The graph test. Plotting against for a direct variation gives a straight line through the origin. The line must pass through , because forces when — no articles cost nothing.
Which is exactly why the taxi fare fails. Its graph is a straight line, but it cuts the vertical axis above the origin at the fixed charge. A relationship of the form with is not a direct variation, however straight its graph is, and this is the single commonest misconception in the chapter.
Written and read * varies directly as *, it means
where is the constant of proportionality, also called the constant of variation.
**Finding needs only one pair of values. Divide the second quantity by the first, and the answer holds for every other pair.
Worked example 1.** If pens cost , find the constant of proportionality and the cost of pens.
Notice what means here — it is the cost of one pen, and it carries the unit rupees per pen. In every direct variation the constant is a rate, which is why identifying it makes the problem concrete.
Worked example 2. A car travels on of fuel. How far will it travel on ?
Everyday pairs that do vary directly:
- Cost and the number of articles bought, at a fixed rate
- Distance and time, at a constant speed
- Weight and the number of identical objects
- The circumference of a circle and its diameter, with
The graph test. Plotting against for a direct variation gives a straight line through the origin. The line must pass through , because forces when — no articles cost nothing.
Which is exactly why the taxi fare fails. Its graph is a straight line, but it cuts the vertical axis above the origin at the fixed charge. A relationship of the form with is not a direct variation, however straight its graph is, and this is the single commonest misconception in the chapter.
How do you solve a direct variation problem?
Two standard methods, and both give the same answer.
The unitary method — find the value for one unit, then multiply.
Worked example 1. If of cloth costs , find the cost of .
The proportion method — set the two ratios equal and cross-multiply.
The same , and the proportion method avoids the intermediate division when it does not come out neatly.
Worked example 2 — working the other way. If of cloth costs , how much cloth can be bought for ?
Or by the unitary method: buys one metre, so buys .
Worked example 3 — a three-step problem. If workers earn in a week, what will workers earn in the same week?
How to set a proportion up safely. Keep like quantities in like positions — both cloth lengths on top and both costs underneath, or both on the same side of each fraction. Mixing a length with a cost across the fraction bar gives an equation that is dimensionally meaningless and an answer that is wrong.
The check that catches a reversed proportion. Ask whether the answer should be bigger or smaller than the given value before calculating. Eight metres is less than fifteen, so the cost must be less than — and is. Had the fractions been inverted the answer would have come to , which fails that test immediately.
The unitary method — find the value for one unit, then multiply.
Worked example 1. If of cloth costs , find the cost of .
The proportion method — set the two ratios equal and cross-multiply.
The same , and the proportion method avoids the intermediate division when it does not come out neatly.
Worked example 2 — working the other way. If of cloth costs , how much cloth can be bought for ?
Or by the unitary method: buys one metre, so buys .
Worked example 3 — a three-step problem. If workers earn in a week, what will workers earn in the same week?
How to set a proportion up safely. Keep like quantities in like positions — both cloth lengths on top and both costs underneath, or both on the same side of each fraction. Mixing a length with a cost across the fraction bar gives an equation that is dimensionally meaningless and an answer that is wrong.
The check that catches a reversed proportion. Ask whether the answer should be bigger or smaller than the given value before calculating. Eight metres is less than fifteen, so the cost must be less than — and is. Had the fractions been inverted the answer would have come to , which fails that test immediately.
How do you test whether a table of values varies directly?
**Divide each by its and check that every ratio is the same. One ratio out of line is enough to disprove it.
Worked example 1 — a table that does vary directly.**
- : , , ,
- : , , ,
Taking each ratio:
Every ratio is , so the quantities do vary directly, with and .
Worked example 2 — a table that does not.
- : , ,
- : , ,
The ratios are not equal, so this is not a direct variation — even though increases steadily as does, and even though the differences are perfectly regular.
Look closely at why it failed. The values rise by each time the values rise by , so the relationship is . That is a straight line, but its constant term means it misses the origin, and the ratio therefore drifts. Regular differences are not the same as a constant ratio, and this pair of examples is the clearest way to see it.
**Completing a table once is known.** With from the first example:
- If , then
- If , then
- If , then
Once the constant is found, every missing entry is a single multiplication or division — which is why finding first is always the right opening move.
A faster test for a longer table. Instead of computing every ratio, check whether ** is the same for the first and last rows**, then spot-check one in the middle. If the relationship really is , any two rows will agree; if it is , the first and last will disagree most.
And a warning about rounding. Real measured data rarely gives ratios that agree exactly. A question with tidy whole numbers expects exact agreement, and a ratio that is nearly equal — like and above — means not a direct variation, not an approximate one.
Worked example 1 — a table that does vary directly.**
- : , , ,
- : , , ,
Taking each ratio:
Every ratio is , so the quantities do vary directly, with and .
Worked example 2 — a table that does not.
- : , ,
- : , ,
The ratios are not equal, so this is not a direct variation — even though increases steadily as does, and even though the differences are perfectly regular.
Look closely at why it failed. The values rise by each time the values rise by , so the relationship is . That is a straight line, but its constant term means it misses the origin, and the ratio therefore drifts. Regular differences are not the same as a constant ratio, and this pair of examples is the clearest way to see it.
**Completing a table once is known.** With from the first example:
- If , then
- If , then
- If , then
Once the constant is found, every missing entry is a single multiplication or division — which is why finding first is always the right opening move.
A faster test for a longer table. Instead of computing every ratio, check whether ** is the same for the first and last rows**, then spot-check one in the middle. If the relationship really is , any two rows will agree; if it is , the first and last will disagree most.
And a warning about rounding. Real measured data rarely gives ratios that agree exactly. A question with tidy whole numbers expects exact agreement, and a ratio that is nearly equal — like and above — means not a direct variation, not an approximate one.
Exam tip
Exam tip: check the ratio, not the direction
To prove direct variation you must show the **ratio is constant**. Saying * increases as increases* proves nothing — does that and is not a direct variation.
Compute every ratio in a table and state that they are equal, giving the value of . To disprove it, one unequal ratio is enough, and you should point to it.
Always find and name the constant with its unit — per pen, per litre. It makes the rest of the problem concrete and it is usually worth a mark.
In a proportion, keep like quantities in like positions. Both lengths on top and both costs underneath.
Predict the direction first. If the new quantity is smaller, the answer must be smaller. That single check catches an inverted proportion instantly.
Remember the graph of a direct variation is a straight line through the origin. A line cutting the axis above the origin is and is not direct variation.
Use the unitary method when the division comes out neatly and proportion when it does not.
And note that equal differences are not the same as a constant ratio — the rows rising by a fixed step is exactly the trap.
Compute every ratio in a table and state that they are equal, giving the value of . To disprove it, one unequal ratio is enough, and you should point to it.
Always find and name the constant with its unit — per pen, per litre. It makes the rest of the problem concrete and it is usually worth a mark.
In a proportion, keep like quantities in like positions. Both lengths on top and both costs underneath.
Predict the direction first. If the new quantity is smaller, the answer must be smaller. That single check catches an inverted proportion instantly.
Remember the graph of a direct variation is a straight line through the origin. A line cutting the axis above the origin is and is not direct variation.
Use the unitary method when the division comes out neatly and proportion when it does not.
And note that equal differences are not the same as a constant ratio — the rows rising by a fixed step is exactly the trap.
Did you know
Why must a proportional graph pass through the origin?
It is easy to treat goes through the origin as an extra rule bolted onto direct variation. It is not a rule at all — it is unavoidable.
Direct variation says . Put into that and you get , whatever the value of . So the point satisfies every direct variation there has ever been, and the line has no choice but to pass through it.
In words: no articles cost nothing, no time covers no distance, no petrol takes you nowhere. Every genuine direct variation makes that statement true, and if a situation makes it false, the situation is not a direct variation.
That is why the taxi fare, the electricity bill with a fixed charge, and the mobile plan with a monthly rental all fail the test. Each one charges something for zero usage, so each one has a line that starts above the origin — and for each one, doubling the usage fails to double the bill. The origin is not a detail of the graph; it is the whole definition, drawn.
Direct variation says . Put into that and you get , whatever the value of . So the point satisfies every direct variation there has ever been, and the line has no choice but to pass through it.
In words: no articles cost nothing, no time covers no distance, no petrol takes you nowhere. Every genuine direct variation makes that statement true, and if a situation makes it false, the situation is not a direct variation.
That is why the taxi fare, the electricity bill with a fixed charge, and the mobile plan with a monthly rental all fail the test. Each one charges something for zero usage, so each one has a line that starts above the origin — and for each one, doubling the usage fails to double the bill. The origin is not a detail of the graph; it is the whole definition, drawn.
Key takeaways
Direct variation: quick revision
- Two quantities are in direct variation when their ratio is constant: means , or .
- is the constant of proportionality, found from one pair of values, and it is always a rate with a unit.
- pens for gives per pen, so pens cost . A car doing on gives , so takes it .
- Examples: cost and quantity, distance and time at constant speed, weight and number of identical objects, circumference and diameter with .
- The graph is a straight line through the origin, because forces at .
- A relationship with — a taxi fare with a fixed charge — is not direct variation, however straight its graph.
- Unitary method: for gives per metre, so costs .
- Proportion: gives ; and buys .
- workers earning means each, so workers earn .
- Keep like quantities in like positions, and predict whether the answer should be bigger or smaller before calculating.
- Testing a table: with gives every ratio , so it varies directly with .
- with gives ratios , and about — not direct variation, since the relationship is .
- Equal differences are not a constant ratio. Once is known, any missing entry is one multiplication or division: with , gives and gives .
Test a few tables of your own and, for each one that fails, work out the it actually follows — seeing where the constant term comes from is what makes the definition stick.
- is the constant of proportionality, found from one pair of values, and it is always a rate with a unit.
- pens for gives per pen, so pens cost . A car doing on gives , so takes it .
- Examples: cost and quantity, distance and time at constant speed, weight and number of identical objects, circumference and diameter with .
- The graph is a straight line through the origin, because forces at .
- A relationship with — a taxi fare with a fixed charge — is not direct variation, however straight its graph.
- Unitary method: for gives per metre, so costs .
- Proportion: gives ; and buys .
- workers earning means each, so workers earn .
- Keep like quantities in like positions, and predict whether the answer should be bigger or smaller before calculating.
- Testing a table: with gives every ratio , so it varies directly with .
- with gives ratios , and about — not direct variation, since the relationship is .
- Equal differences are not a constant ratio. Once is known, any missing entry is one multiplication or division: with , gives and gives .
Test a few tables of your own and, for each one that fails, work out the it actually follows — seeing where the constant term comes from is what makes the definition stick.